Long-range topological insulators and weakened bulk-boundary correspondence

We formalize the appearance of new types of insulators in long-range (LR) fermionic systems. These phases are not included in the "ten-fold way classification" (TWC) for the short-range (SR) topological insulators. This conclusion is obtained studying at first specific one-dimensional LR examples, in particular their phase diagrams and contents in symmetries and entanglement. The purely long-range phases (LRP) are signaled by the violation of the area-law for the Von Neumann entropy and by corresponding peculiar distributions for the entanglement spectrum (ES). The origin of the deviations from the TWC is analyzed from a more general point of view and in any dimension. In particular, it is found related with a particular type of divergences occurring in the spectrum, due to the LR couplings. A satisfying characterization for the LRP can be achieved at least for one-dimensional systems, as well as the connected definition of a nontrivial topology, provided a careful evaluation of the LR contributions. Our results lead to reconsider the definition of correlation length in LR systems. The same analysis also allows to infer, at least for one-dimensional models, the weakening of the bulk-boundary correspondence, due to the important correlations between bulk and edges, and consequently to clarify the nature of the massive edge states appearing in the topological LR. The emergence of this peculiar edge structure is signaled by the bulk ES. The stability of the LRP against finite-size effects, relevant in current experiments, and against local disorder is discussed, showing that the latter ingredient can even strengthen the effect of the LR couplings. Finally, we analyze the entanglement content of the paradigmatic LR Ising spin chain, inferring again important deviations from the SR regime, and the limitations of bulk-boundary (tensor-network based) approaches to classify LR spin models.